Boundaries of Groups and Kleinian Groups — Problem 35

Definitions and mathematical statement

For a homeomorphism $f:\mathbb R^n\to\mathbb R^n$, put \[ H_f(x)=\limsup_{r\downarrow0} \frac{\max_{|y-x|=r}|f(y)-f(x)|} {\min_{|y-x|=r}|f(y)-f(x)|}. \] It is quasiconformal in the metric sense when $\sup_x H_f(x)<\infty$. A smooth diffeomorphism is a bijective smooth map with smooth inverse. Approximation is in the compact-open topology: $f_j\to f$ uniformly on every compact subset of the domain.

Is every quasiconformal homeomorphism $f:\mathbb R^4\to\mathbb R^4$ a compact-open limit of smooth diffeomorphisms $f_j:\mathbb R^4\to\mathbb R^4$? Equivalently, for each compact $K\subset\mathbb R^4$ and each $\varepsilon>0$, can one find a global smooth diffeomorphism with \[ \sup_{x\in K}|f_j(x)-f(x)|<\varepsilon? \] The source states the question in all dimensions and records affirmative approximation results away from dimension four. This entry therefore isolates that exceptional dimension. The approximating maps in this question need not have a uniform quasiconformal distortion bound; the uniform-distortion requirement is a separate stronger problem.

Global bijectivity is essential. Convolution can approximate coordinate functions smoothly on compact sets, but does not automatically preserve injectivity or produce a smooth inverse on all of Euclidean space. Likewise, approximation on an arbitrary pair of domains is a different formulation from the whole-space problem stated here. The dimension remains fixed throughout the approximation.

Short English statement

Can every quasiconformal homeomorphism of four-dimensional Euclidean space be approximated by global smooth diffeomorphisms?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200035 (AMR-061-0035), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 35, PDF page 11. Original source indexed by AMR-061-0035. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.